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ON THE COLOURS  OF  THICK  PLATES.                     179
luminous point is seen distinctly. Omitting further consideration of the conditions of distinctness, let us regard the eye as a point, and investigate the form and character of the rings.
19. Let /, g, h be the co-ordinates of the eye. To find the retardation, it will be sufficient, as in Art. 9, to write f, g, h for a!, V, c, and take x, y to denote the co-ordinates of that point of the mirror on which a given point of a ring is seen projected. The whole retardation is the sum of the expressions in (5) and (11); and making the above substitution we find
Hence the bands still form a system of concentric circles.    If F, F be the co-ordinates of the centre of the system,
Y -
.             .
\h     cj \h    c     pj and F may be written down from symmetry.
The equations of a line joining the eye and the luminous point are
£ — a _ -i? — 6_ f -^c f—ct    g — b    k — c'
At the point in which this line cuts the mirror £= 0, or at least is a very small quantity, which may be neglected. Hence we have
from whence 77 may be written down if required. If ^15 rjl denote the co-ordinates of the point in which the line joining the eye and the image meets the mirror, ^, ^ may be obtained from £, q by writing a1? blt cl for a, b} c, where ap bl9 c^ denote the co-ordinates of the image. Observing that
ai =. _ a          l = 2 _ -
ol         c '        cl    p    c '